void MeshRefinement::flag_elements_by_mean_stddev (const ErrorVector & error_per_cell,
const Real refine_frac,
const Real coarsen_frac,
const unsigned int max_l)
{
// The function arguments are currently just there for
// backwards_compatibility
if (!_use_member_parameters)
{
// If the user used non-default parameters, lets warn
// that they're deprecated
if (refine_frac != 0.3 ||
coarsen_frac != 0.0 ||
max_l != libMesh::invalid_uint)
libmesh_deprecated();
_refine_fraction = refine_frac;
_coarsen_fraction = coarsen_frac;
_max_h_level = max_l;
}
// Get the mean value from the error vector
const Real mean = error_per_cell.mean();
// Get the standard deviation. This equals the
// square-root of the variance
const Real stddev = std::sqrt (error_per_cell.variance());
// Check for valid fractions
libmesh_assert_greater_equal (_refine_fraction, 0);
libmesh_assert_less_equal (_refine_fraction, 1);
libmesh_assert_greater_equal (_coarsen_fraction, 0);
libmesh_assert_less_equal (_coarsen_fraction, 1);
// The refine and coarsen cutoff
const Real refine_cutoff = mean + _refine_fraction * stddev;
const Real coarsen_cutoff = std::max(mean - _coarsen_fraction * stddev, 0.);
// Loop over the elements and flag them for coarsening or
// refinement based on the element error
for (auto & elem : _mesh.active_element_ptr_range())
{
const dof_id_type id = elem->id();
libmesh_assert_less (id, error_per_cell.size());
const ErrorVectorReal elem_error = error_per_cell[id];
// Possibly flag the element for coarsening ...
if (elem_error <= coarsen_cutoff)
elem->set_refinement_flag(Elem::COARSEN);
// ... or refinement
if ((elem_error >= refine_cutoff) && (elem->level() < _max_h_level))
elem->set_refinement_flag(Elem::REFINE);
}
}
int main(int argc, char** argv)
{
// Initialize libMesh.
LibMeshInit init (argc, argv);
// Adaptive constraint calculations for fine Hermite elements seems
// to require half-decent precision
#ifdef LIBMESH_DEFAULT_SINGLE_PRECISION
libmesh_example_assert(false, "double precision");
#endif
// This example requires Adaptive Mesh Refinement support
#ifndef LIBMESH_ENABLE_AMR
libmesh_example_assert(false, "--enable-amr");
#else
// This example requires second derivative calculation support
#ifndef LIBMESH_ENABLE_SECOND_DERIVATIVES
libmesh_example_assert(false, "--enable-second");
#else
// Parse the input file
GetPot input_file("adaptivity_ex4.in");
// Read in parameters from the input file
const unsigned int max_r_level = input_file("max_r_level", 10);
const unsigned int max_r_steps = input_file("max_r_steps", 4);
const std::string approx_type = input_file("approx_type",
"HERMITE");
const unsigned int uniform_refine =
input_file("uniform_refine", 0);
const Real refine_percentage =
input_file("refine_percentage", 0.5);
const Real coarsen_percentage =
input_file("coarsen_percentage", 0.5);
const unsigned int dim =
input_file("dimension", 2);
const unsigned int max_linear_iterations =
input_file("max_linear_iterations", 10000);
// Skip higher-dimensional examples on a lower-dimensional libMesh build
libmesh_example_assert(dim <= LIBMESH_DIM, "2D/3D support");
// We have only defined 2 and 3 dimensional problems
libmesh_assert (dim == 2 || dim == 3);
// Currently only the Hermite cubics give a 3D C^1 basis
libmesh_assert (dim == 2 || approx_type == "HERMITE");
// Create a mesh.
Mesh mesh;
// Output file for plotting the error
std::string output_file = "";
if (dim == 2)
output_file += "2D_";
else if (dim == 3)
output_file += "3D_";
if (approx_type == "HERMITE")
output_file += "hermite_";
else if (approx_type == "SECOND")
output_file += "reducedclough_";
else
output_file += "clough_";
if (uniform_refine == 0)
output_file += "adaptive";
else
output_file += "uniform";
std::string exd_file = output_file;
exd_file += ".e";
output_file += ".m";
std::ofstream out (output_file.c_str());
out << "% dofs L2-error H1-error H2-error\n"
<< "e = [\n";
// Set up the dimension-dependent coarse mesh and solution
// We build more than one cell so as to avoid bugs on fewer than
// 4 processors in 2D or 8 in 3D.
if (dim == 2)
{
MeshTools::Generation::build_square(mesh, 2, 2);
exact_solution = &exact_2D_solution;
exact_derivative = &exact_2D_derivative;
exact_hessian = &exact_2D_hessian;
forcing_function = &forcing_function_2D;
}
else if (dim == 3)
{
MeshTools::Generation::build_cube(mesh, 2, 2, 2);
exact_solution = &exact_3D_solution;
exact_derivative = &exact_3D_derivative;
exact_hessian = &exact_3D_hessian;
forcing_function = &forcing_function_3D;
}
// Convert the mesh to second order: necessary for computing with
// Clough-Tocher elements, useful for getting slightly less
// broken visualization output with Hermite elements
mesh.all_second_order();
// Convert it to triangles if necessary
if (approx_type != "HERMITE")
MeshTools::Modification::all_tri(mesh);
// Mesh Refinement object
MeshRefinement mesh_refinement(mesh);
mesh_refinement.refine_fraction() = refine_percentage;
mesh_refinement.coarsen_fraction() = coarsen_percentage;
mesh_refinement.max_h_level() = max_r_level;
// Create an equation systems object.
EquationSystems equation_systems (mesh);
// Declare the system and its variables.
// Create a system named "Biharmonic"
LinearImplicitSystem& system =
equation_systems.add_system<LinearImplicitSystem> ("Biharmonic");
// Adds the variable "u" to "Biharmonic". "u"
// will be approximated using Hermite tensor product squares
// or (possibly reduced) cubic Clough-Tocher triangles
if (approx_type == "HERMITE")
system.add_variable("u", THIRD, HERMITE);
else if (approx_type == "SECOND")
system.add_variable("u", SECOND, CLOUGH);
else if (approx_type == "CLOUGH")
system.add_variable("u", THIRD, CLOUGH);
else
libmesh_error();
// Give the system a pointer to the matrix assembly
// function.
system.attach_assemble_function (assemble_biharmonic);
// Initialize the data structures for the equation system.
equation_systems.init();
// Set linear solver max iterations
equation_systems.parameters.set<unsigned int>
("linear solver maximum iterations") = max_linear_iterations;
// Linear solver tolerance.
equation_systems.parameters.set<Real>
("linear solver tolerance") = TOLERANCE * TOLERANCE;
// Prints information about the system to the screen.
equation_systems.print_info();
// Construct ExactSolution object and attach function to compute exact solution
ExactSolution exact_sol(equation_systems);
exact_sol.attach_exact_value(exact_solution);
exact_sol.attach_exact_deriv(exact_derivative);
exact_sol.attach_exact_hessian(exact_hessian);
// Construct zero solution object, useful for computing solution norms
// Attaching "zero_solution" functions is unnecessary
ExactSolution zero_sol(equation_systems);
// A refinement loop.
for (unsigned int r_step=0; r_step<max_r_steps; r_step++)
{
mesh.print_info();
equation_systems.print_info();
std::cout << "Beginning Solve " << r_step << std::endl;
// Solve the system "Biharmonic", just like example 2.
system.solve();
std::cout << "Linear solver converged at step: "
<< system.n_linear_iterations()
<< ", final residual: "
<< system.final_linear_residual()
<< std::endl;
// Compute the error.
exact_sol.compute_error("Biharmonic", "u");
// Compute the norm.
zero_sol.compute_error("Biharmonic", "u");
// Print out the error values
std::cout << "L2-Norm is: "
<< zero_sol.l2_error("Biharmonic", "u")
<< std::endl;
std::cout << "H1-Norm is: "
<< zero_sol.h1_error("Biharmonic", "u")
<< std::endl;
std::cout << "H2-Norm is: "
<< zero_sol.h2_error("Biharmonic", "u")
<< std::endl
<< std::endl;
std::cout << "L2-Error is: "
<< exact_sol.l2_error("Biharmonic", "u")
<< std::endl;
std::cout << "H1-Error is: "
<< exact_sol.h1_error("Biharmonic", "u")
<< std::endl;
std::cout << "H2-Error is: "
<< exact_sol.h2_error("Biharmonic", "u")
<< std::endl
<< std::endl;
// Print to output file
out << equation_systems.n_active_dofs() << " "
<< exact_sol.l2_error("Biharmonic", "u") << " "
<< exact_sol.h1_error("Biharmonic", "u") << " "
<< exact_sol.h2_error("Biharmonic", "u") << std::endl;
// Possibly refine the mesh
if (r_step+1 != max_r_steps)
{
std::cout << " Refining the mesh..." << std::endl;
if (uniform_refine == 0)
{
ErrorVector error;
LaplacianErrorEstimator error_estimator;
error_estimator.estimate_error(system, error);
mesh_refinement.flag_elements_by_elem_fraction (error);
std::cout << "Mean Error: " << error.mean() <<
std::endl;
std::cout << "Error Variance: " << error.variance() <<
std::endl;
mesh_refinement.refine_and_coarsen_elements();
}
else
{
mesh_refinement.uniformly_refine(1);
}
// This call reinitializes the \p EquationSystems object for
// the newly refined mesh. One of the steps in the
// reinitialization is projecting the \p solution,
// \p old_solution, etc... vectors from the old mesh to
// the current one.
equation_systems.reinit ();
}
}
#ifdef LIBMESH_HAVE_EXODUS_API
// Write out the solution
// After solving the system write the solution
// to a ExodusII-formatted plot file.
ExodusII_IO (mesh).write_equation_systems (exd_file,
equation_systems);
#endif // #ifdef LIBMESH_HAVE_EXODUS_API
// Close up the output file.
out << "];\n"
<< "hold on\n"
<< "loglog(e(:,1), e(:,2), 'bo-');\n"
<< "loglog(e(:,1), e(:,3), 'ro-');\n"
<< "loglog(e(:,1), e(:,4), 'go-');\n"
<< "xlabel('log(dofs)');\n"
<< "ylabel('log(error)');\n"
<< "title('C1 " << approx_type << " elements');\n"
<< "legend('L2-error', 'H1-error', 'H2-error');\n";
// All done.
return 0;
#endif // #ifndef LIBMESH_ENABLE_SECOND_DERIVATIVES
#endif // #ifndef LIBMESH_ENABLE_AMR
}